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Local Solutions of Fully Nonlinear
Weakly Hyperbolic Differential Equations
in Sobolev spaces
M. Dreher and M. Reissig
Abstract
The goal of the present paper is to study fully nonlinear weakly hyperbolicequations of second order with space– and time degeneracy. A local existenceresult in Sobolev spaces under sharp Levi conditions of C∞ type is proved.These Levi conditions and the behaviour of the nonlinearities determine therequired smoothness of the data and the loss of Sobolev regularity.
Key words: fully nonlinear weakly hyperbolic equations, local solutions inSobolev spaces, energy method
AMS Classification: 35L70, 35L80
Introduction
In this paper we want to study fully nonlinear weakly hyperbolic differential equa-tions in one space dimension with time– and spatial degeneracy. We will prove alocal existence result in Sobolev spaces for the Cauchy problem
F (utt, σ(x)λ(t)uxt, σ(x)2λ(t)2uxx, σ(x)λ′(t)ux, ut, u, x, t) = 0, (0.1)
u(x, 0) = ϕ0(x), ut(x, 0) = ϕ1(x). (0.2)
We assume that this Cauchy problem is strictly hyperbolic if σ(x) ≡ 1 and λ(t) ≡ 1.The functions σ(x) and λ(t) describe the degeneration, which occurs for σ(x) = 0(spatial degeneracy) and λ(t) = 0 (time degeneracy). The first question is that forclasses of well–posedness with respect to x. If we restrict ourselves to Gevrey classesof order ≤ 2, then we can use ideas of [Kaj83] to prove a local existence result for
F (utt, σ(x)λ(t)uxt, σ(x)2λ(t)2uxx, ux, ut, u, x, t) = 0,
u(x, 0) = ϕ0(x), ut(x, 0) = ϕ1(x).
To overcome the critical order 2 we need so–called Levi conditions. In [RY96] a localexistence result in all Gevrey spaces could be proved under special assumptions for
utt − (a(x, t)ux)x = f(x, t, u, ux)
2
where the nonlinear Levi condition of C∞–type
∣
∣
∣∂l
pf(x, t, u, p)∣
∣
∣≤ CKM ls
K l!s′√
a(x, t) (0.3)
is satisfied for all l ≥ 1 and all compact sets K ⊂ Rx × [0, T ] × Ru × Rp. There aredifferent results for local existence in C∞ for special quasilinear weakly hyperbolicmodel equations under quite different assumptions [D’A93], [DT95], [Man96]. Butin all these model equations the nonlinearities depend at most on u and ut. In thecase of spatial degeneracy (λ′(t) and λ(t) are absent in (0.1)), the C∞– and Gevreywell posedness is proved in [DR]. What about results for local existence of Sobolevsolutions? In [Ner66] one can find one of the first results for the time degeneracycase
utt − λ(t)2uxx − a(x, t)ux − b(x, t)ut − c(x, t)u = f(x, t),
u(x, 0) = ϕ0(x), ut(x, 0) = ϕ1(x)
under the Levi condition
lim supt→+0
|a(x, t)|λ′(t)
≤ q < ∞.
The loss of regularity depends on q and the Levi condition is sharp. Later weaklyhyperbolic Cauchy problems of the form
utt −N∑
i,j=1
(aij(x, t)uxi)xj
+N∑
b=1
bi(x, t)uxi+ b0(x, t)ut + c(x, t)u = f(x, t),
u(x, 0) = ϕ0(x), ut(x, 0) = ϕ1(x)
were studied in [Ole70] under the Levi condition
ct
(
N∑
i=1
bi(x, t)ξi
)2
≤ AN∑
i,j=1
aij(x, t)ξiξj + ∂t
N∑
i,j=1
aij(x, t)ξiξj
.
In the case of time degeneracy this Levi condition is only sharp if we have a degen-eracy of finite order (compare with the Levi condition of Nersesian). In [Rei96] itwas shown for the model problem
utt − λ(t)2 4 u = f(x, t, u, ut, λ′(t)∇xu),
u(x, 0) = ϕ0(x), ut(x, 0) = ϕ1(x)
how to prove the local existence of Sobolev solutions. The difficulty is to showhow to overcome the quasilinear structure with the loss of Sobolev regularity whichappears even in the linear theory. In [KY] a local existence result could be derivedfor a general quasilinear weakly hyperbolic equation of higher order in the case
3
of time degeneracy. To prove local existence of Sobolev solutions we need C∞–type Levi conditions. The special structure of the arguments in (0.1) ensures thefulfilment of these conditions. There are other ways to describe Levi conditions.The investigations in this paper serve as a preparation for further studies which willbe devoted to the question for global existence. As usual, in those Cauchy problemsthe function F depends only on the solution and some of its derivatives. In this caseit is necessary to use the functions σ and λ to formulate the C∞–Levi conditions.The study of a model equation
utt − σ(x)2λ(t)2 4 u = f(∇xu)
under the assumption
|∂pf(p)| ≤ Cσ(x)λ′(t)
(similar to (0.3)) leads to spaces of solutions with asymptotics, which seem to beextremely difficult to handle.
The aim of this paper is to prove
Theorem 0.1 We suppose:
A 1 The function F (u11, σλu12, σ2λ2u22, σλ′u2, u1, u, x, t) is defined on the set M ×
P × I, where M is an open set in R6, P ⊂ R is a compact interval and
I = [0, T ].
A 2 The functions F , σ, ϕ0, ϕ1 are P–periodic in x.
A 3 The function λ ∈ C3([0, T ]) fulfils the conditions
λ(0) = λ′(0) = 0, λ(t) > 0, λ′(t) > 0 (t > 0). (0.4)
A 4 It is assumed that σ ∈ HN+2per (P ), ϕ0 ∈ HN+1
per (P ), ϕ1 ∈ HNper(P ), where N ≥ 5
and HNper(P ) denotes the functions from HN
loc(R) which are P–periodic.
A 5 The derivatives Fu11 ,. . .Ft belong to C1([0, T ], C∞(R6) × HNper(P )).
A 6 With a suitable constant α it holds |Fu11 | ≥ α > 0 on M × P × I.
A 7 Let on M × P × I be
(
Fσλu12
Fu11
)2
− 4Fσ2λ2u22
Fu11
≥ γ > 0.
From condition A 6 we conclude that the set in M × P × I which is given by F = 0can be represented in the form
u11 = G(σλu12, σ2λ2u22, σλ′u2, u1, u, x, t).
4 1 EVOLUTION OPERATORS AND ENERGY ESTIMATES
A 8 Let ϕ2(x) be the function which is defined by
ϕ2(x) = G(0, 0, 0, ϕ1(x), ϕ0(x), x, 0).
We assume that the set
K := {(ϕ2(x), 0, 0, 0, ϕ1(x), ϕ0(x), x, 0) : x ∈ P}
is contained in M × P × {0}.
Then there exist constants r ∈ N and T ∗ ∈ (0, T ] such that the Cauchy prob-lem (0.1), (0.2) has a solution u,
u, ut, σλux ∈ C2(
[0, T ],HN−rper (P )
)
if N −r ≥ 5. The constant r describes the loss of Sobolev regularity and may dependon N . One can show that N − r ≥ 5 for sufficiently large N . This guarantees theexistence of a solution for large N .
For the convenience of the reader we give an overview of this paper: In the firstsection we provide some tools which will be used in later sections. The second andthird section deal with problems which have no time degeneracy, whereas the fourthand fifth section include this degeneration.
We study linear equations in Section 2. At first we derive energy estimates. Theexistence of the solution is proved by applying a smoothing technique and the ab-stract Theorem of Cauchy–Kowalewskaja. We can prove convergence of a suitablesequence of solutions by our energy estimates.
Quasilinear equations are studied in Section 3. We prove the existence of a solutionby linearization and standard iteration. The convergence of this sequence is againshown by energy estimates. Although we consider only scalar equations it is obviousthat similar results can be proved for quasilinear systems with diagonal principalpart.
In Section 4 we consider quasilinear equations with both degeneracies. We approx-imate the problem by problems without time degeneracy and apply the results ofSection 3.
We show how one can reduce fully nonlinear equations to a quasilinear system withdiagonal principal part in Section 5.
In Section 6 we study some examples. We can show that the loss of Sobolev regularitypredicted in Section 4 occurs, indeed. Additionally we give some remarks which areof independent interest.
1 Evolution Operators and Energy Estimates
In this section we assemble some tools which we need in later proofs.
5
We define the partial energies
ej(u)(t) :=
(∫
P
(∂jxu(x, t))2 dx
)12
and the energies of finite order
EN (u)(t) :=
N∑
j=0
ej(u)(t).
It is worth to remark that EN (u)(t) = ‖u(t)‖HN (P ). These energies have the prop-erties
EN (uv)(t) ≤ Cprod,NEN (u)(t)EN (v)(t), N ≥ 1,
E0(uv)(t) ≤ Cprod,0E1(u)(t)E0(v)(t),
‖∂xu(x, t)‖∞ ≤ CP e2(u)(t), u ∈ H2per(P ).
Proposition 1.1 Let N ≥ 1, N ′ = max(2, N) and
λ ∈ C(
[0, T ],HN ′
per(P ))
,
f ∈ C(
[0, T ],HN−1per (P )
)
∩ L1(
[0, T ],HNper(P )
)
.
Let u(x, t) be a solution of
ut(x, t) − λ(x, t)ux(x, t) = f(x, t), (x, t) ∈ P × [0, T ].
(a) If u ∈ L∞(
[0, T ],HN+1per (P )
)
and ut ∈ L∞(
[0, T ],HNper(P )
)
, then
EN (u)′(t) ≤ CNEN (u)(t) + EN (f)(t). (1.1)
(b) If u ∈ L∞(
[0, T ],HNper(P )
)
and ut ∈ L∞(
[0, T ],HN−1per (P )
)
, then
EN−1(u)′(t) ≤ CN−1EN−1(u)(t) + EN−1(f)(t) (1.2)
and if 0 ≤ t0 ≤ t ≤ T , then it holds
EN (u)(t) ≤ EN (u)(t0)eCN (t−t0) +
∫ t
t0
eCN (t−τ)EN (f)(τ) dτ. (1.3)
The constants Cn depend on ‖λ‖C
“
[0,T ],Hmax(2,n)per (P )
”.
Proof of (a) Let 0 ≤ j ≤ N . Then it holds
ej(u)(t)ej(u)′(t) =
∫
P
(
∂jxu) (
∂jxut
)
dx =
∫
P
(
∂jxu)
∂jx (f + λux) dx
≤ ej(u)(t)ej(f)(t) +
∫
P
(
∂jxu)
λ(
∂j+1x u
)
dx
+
j∑
n=1
(
j
n
)∫
P
(
∂jxu)
(∂nxλ)
(
∂j−n+1x u
)
dx.
6 1 EVOLUTION OPERATORS AND ENERGY ESTIMATES
The second integral of the right–hand side can be estimated after partial integrationby ‖λx‖∞ ej(u)2 ≤ C ‖λ‖HN′ (P ) ej(u)2. We get for n = j in the last integral theinequality
∫
P
(
∂jxu) (
∂jxλ)
(∂xu) dx ≤ ej(u) ‖λ‖Hj(P ) ‖ux‖∞ ≤ C ‖λ‖Hj(P ) ej(u)e2(u),
and for n ≤ j − 1
∫
P
(
∂jxu)
(∂nxλ)
(
∂j−n+1x u
)
dx ≤ ej(u) ‖λ‖Cn ej−n+1(u)
≤ C ‖λ‖HN′ (P ) ej(u)ej−n+1(u).
Summation (j = 0, . . . , N) yields the assertion.
Proof of (b) It remains to prove the estimate (1.3). We approximate
Aper(P ) 3 un(t0)HN
per(P )−−−−−→ u(t0),
C ([0, T ],Aper(P )) 3 fnL1([0,T ],HN
per(P ))−−−−−−−−−−−→ f,
C ([0, T ],Aper(P )) 3 λnC
“
[0,T ],HN′
per(P )”
−−−−−−−−−−−→ λ.
Here we denote by Aper(P ) the space of functions which are analytic and P–periodic.
Lemma 4.1 from [DR] shows that there exists a solution un ∈ C1 ([t0, T ],Aper(P ))of
unt − λnun
x = fn.
By Gronwall’s Lemma and (a) we obtain for every n
EN (un)(t) ≤ EN (un)(t0)eCN (t−t0) +
∫ t
t0
eCN (t−τ)EN (fn)(τ) dτ.
From (a) and
(u − un)t − λ(u − un)x = f − fn − (λn − λ)unx
we conclude that ‖un − u‖HN−1(P ) → 0.
Furthermore, there exists a constant C with ‖un(t)‖HN (P ) ≤ C for all n, t. We fix
t ≥ t0. There exists a subsequence un∗ (t) weakly converging in HN
per(P ) to somefunction w(t). The uniqueness of the limit yields u(t) = w(t). Hence
EN (u)(t) ≤ lim infn→∞
EN (un)(t)
≤ lim infn→∞
(
EN (un)(t0)eCN (t−t0) +
∫ t
t0
eCN (t−τ)EN (fn)(τ) dτ
)
.
From this and the approximations we have (1.3). �
7
We will use these results to estimate the solutions of the weakly hyperbolic Cauchyproblem
utt(x, t) + σ(x)b(x, t)uxt(x, t) − σ2(x)a(x, t)uxx(x, t) = f(x, t), (1.4)
u(x, 0) = u0(x), ut(x, 0) = u1(x), (1.5)
b2(x, t) + 4a(x, t) ≥ γ > 0, (x, t) ∈ P × [0, T ]. (1.6)
We factorize the differential operator:
l1,2(x, t) := σ(x)β1,2(x, t) :=1
2σ(x)
(
−b(x, t) ±√
b2(x, t) + 4a(x, t))
,
∂1,2 := ∂t − l1,2(x, t)∂x,
∂1∂2u = utt + σbuxt − σ2auxx + (l1l2,x − l2,t)ux
and define the energy
EN (u)(t) := EN (u)(t) + EN (∂1u)(t) + EN (∂2u)(t).
Hence we obtain
(∂1∂2 + A1(∂1 − ∂2)) u = f, (1.7)
(∂2∂1 + A2(∂2 − ∂1)) u = f, (1.8)
A1(x, t) =l1l2,x − l2,t
l1 − l2, A2(x, t) =
l2l1,x − l1,t
l1 − l2.
This factorization is the essential idea of the proof of the following
Proposition 1.2 We assume with N ≥ 2
σ ∈ HNper(P ), (1.9)
a, b ∈ C1(
[0, T ],HNper(P )
)
, (1.10)
f ∈ C(
[0, T ],HNper(P )
)
. (1.11)
Let u be a solution of (1.4) with u, ∂1u, ∂2u ∈ C(
[0, T ],HMper(P )
)
.
(a) If 0 ≤ M ≤ N − 1, then
EM (u)(t) ≤ (1.12)
≤ eDM (t−t0)
(
EM (u)(t0)eCM (t−t0) + 2
∫ t
t0
eCM (t−τ)EM (f)(τ) dτ
)
.
(b) If we assume additionally σa, σb ∈ C(
[0, T ],HN+1per (P )
)
and σ ∈ HN+1per (P ),
then the estimate (1.12) holds for M = N , too.
The constants DM , CM depend on
EM ′(Aj), ‖lj‖C([0,T ],HM′
per(P )) , j = 1, 2, M ′ = max(M, 2).
8 1 EVOLUTION OPERATORS AND ENERGY ESTIMATES
Proof The assumptions guarantee Aj ∈ C(
[0, T ],HN−1per (P )
)
. The application ofProposition 1.1 to
∂1∂2u = f − A1(∂1 − ∂2)u
shows that
EM (∂2u)(t) ≤EM (∂2u)(t0)eCM (t−t0) +
∫ t
t0
eCM (t−τ)EM (f − A1(∂1 − ∂2)u)(τ) dτ.
We can derive a similar estimate for EM (∂1u)(t). Furthermore,
EM (u)(t) ≤ EM (u)(t0)eCM (t−t0) +
∫ t
t0
eCM (t−τ)EM (∂1u)(τ) dτ.
We haveEM (Ai∂ju) ≤ C ′
MEM ′(Ai)EM (∂ju),
hence
EM (u)(t) ≤EM (u)(t0)eCM (t−t0) + 2
∫ t
t0
eCM (t−τ)EM (f)(τ) dτ
+ C ′′M
∫ t
t0
eCM (t−τ)EM (u)(τ) dτ.
By Gronwall’s Lemma, it follows (1.12).
To prove (b), we only note that Aj ∈ C(
[0, T ],HNper(P )
)
. �
Remark 1.1 We can prove a similar result for systems
ui,tt(x, t) + σbui,xt(x, t) − σ2aui,xx(x, t) = fi(x, t), i = 1, . . . , n
and the energy EM (~u)(t) =∑n
i=1 EM (ui)(t). This is possible since the system hasdiagonal structure.
The next lemma will be needed in Section 4. We define the energy
FN (u)(t) :=
N∑
j=0
(∫
P
∣
∣∂jxu(x, t)
∣
∣
2+∣
∣∂jxut(x, t)
∣
∣
2dx
) 12
=
N∑
j=0
fj(t).
Lemma 1.1 Let u ∈ C2(
[0, T ],HNper(P )
)
be a solution of the following ordinarydifferential equation with parameter
utt + h1(x, t)ut + h2(x, t)u = g(x, t),
where h1, h2, g ∈ L∞(
[0, T ],HNper(P )
)
.
Then it holds
FN (u)(t) ≤ FN (u)(0)eCt +
∫ t
0eC(t−τ)EN (g)(τ) dτ.
9
Proof From the inequalities
fj(u)′(t)fj(u)(t) =
∫
P
((
∂jxu) (
∂jxut
)
+(
∂jxut
) (
∂jxutt
))
dx
≤ fj(u)(t)2 +
∫
P
(
∂jxut
) (
∂jx (g − h1ut − h2u)
)
dx
≤ fj(u)(t)2 + fj(u)(t) (fj(g)(t) + CFN (u)(t) (FN (h1)(t) + FN (h2)(t)))
we deduce that
FN (u)′(t) ≤ EN (g)(t) + (1 + C (EN (h1)(t) + EN (h2)(t)))FN (u)(t).
Gronwall’s Inequality implies the assertion. �
We will need the following generalization of the well–known Gronwall’s Lemma.
Lemma 1.2 (Nersesjan) Let y(t) ∈ C([0, T ]) ∩ C1(0, T ) be a solution of the dif-ferential inequality
y′(t) ≤ K(t)y(t) + f(t), 0 < t < T,
where the functions K(t) and f(t) belong to C(0, T ). We assume for every t ∈ (0, T )and every δ ∈ (0, t)
∫ δ
0K(τ) dτ = ∞,
∫ T
δ
K(τ) dτ < ∞,
limδ→+0
∫ t
δ
exp
(∫ t
s
K(τ) dτ
)
f(s) ds exists,
limδ→+0
y(δ) exp
(∫ t
s
K(τ) dτ
)
= 0.
Then it holds
y(t) ≤∫ t
0exp
(∫ t
s
K(τ) dτ
)
f(s) ds.
2 Existence Results for Linear Equations with Spatial
Degeneracy
Lemma 2.1 We assume N ≥ 3, (1.6), (1.9), (1.10), (1.11) and
u0 ∈ HN+1per (P ), u1 ∈ HN
per(P ). (2.1)
Then the weakly hyperbolic Cauchy problem (1.4), (1.5) has a uniquely deter-mined solution u ∈ C1
(
[0, T ],HN−1per (P )
)
with ∂ju ∈ C(
[0, T ],HN−1per (P )
)
and
utt ∈ C(
[0, T ],HN−2per (P )
)
.
10 2 EX. RESULTS FOR LIN. EQQ. WITH SPATIAL DEGENERACY
Proof We approximate
C1 ([0, T ],Aper(P )) 3 an, bnC1([0,T ],HN
per(P ))−−−−−−−−−−−→ a, b,
Aper(P ) 3 σnHN
per(P )−−−−−→ σ,
C ([0, T ],Aper(P )) 3 fnC([0,T ],HN
per(P ))−−−−−−−−−−−→ f,
Aper(P ) 3 un0
HN+1per (P )−−−−−−→ u0,
Aper(P ) 3 un1
HNper(P )
−−−−−→ u1.
From Theorem 4.1 of [DR] we deduce that the Cauchy problem
utt + σnbnuxt − (σn)2uxx = fn, (2.2)
u(x, 0) = un0 (x), ut(x, 0) = un
1 (x)
has a solution un ∈ C2 ([0, T ],Aper(P )). The norms∥
∥
∥lnj
∥
∥
∥
C([0,T ],HNper(P ))
and∥
∥
∥Anj
∥
∥
∥
C([0,T ],HN−1per (P ))
are uniformly bounded with respect to n. The Proposition 1.2
givesEn
N−1(un)(t) ≤ C, ∀n, t. (2.3)
We show that (un) is a Cauchy–sequence in suitable Banach spaces. Obviously,
(un − um)tt + σnbn(un − um)xt − (σn)2an(un − um)xx
= fn − fm − (σnbn − σmbm) umxt + (σnan − σmam) um
xx. (2.4)
Without loss of generality we may assume that
‖an − a‖ ≤ 1
n, . . . , ‖un
1 − u1‖ ≤ 1
n.
From this it follows that we can estimate the L2(P )–Norm of the right–hand sideof (2.4) by C( 1
n+ 1
m). Here we used N ≥ 3 and the uniform estimates (2.3).
Proposition 1.2 leads to
En0 (un − um)(t) ≤ C
(
1
n+
1
m
)
. (2.5)
Nirenberg–Gagliardo interpolation and (2.3), (2.5) imply
EnN−2(u
n − um)(t) ≤ C
(
1
n+
1
m
)θ
, 0 < θ < 1.
Consequently, there exists a function u ∈ C(
[0, T ],HN−2per (P )
)
with ∂ju ∈C(
[0, T ],HN−2per (P )
)
and
(un, ∂n1 un, ∂n
2 un)C([0,T ],HN−2
per (P ))−−−−−−−−−−−−→ (u, ∂1u, ∂2u).
11
It is easy to show that u is a solution of (1.4), (1.5).
Now we show the better regularity of u. We fix t0 ∈ [0, T ]. From ‖un(t0)‖HN−1(P ) ≤C we gain the existence of a subsequence un
∗ (t0) with un∗ (t0) ⇀ wt0 in HN−1
per (P ). The
embedding of the dual spaces (HN−2per (P ))′ ⊂ (HN−1
per (P ))′ is continuous and dense,
hence un∗ (t0) ⇀ wt0 in HN−2
per (P ). On the other hand, we have un(t0) → u(t0) in
HN−2per (P ). This yields the weak convergence of the whole sequence and wt0 = u(t0),
hence u ∈ L∞(
[0, T ],HN−1per (P )
)
(we do not study the question whether u is Bochner–measurable). Similar arguments apply to ∂ju. It follows that
u(t) ⇀ u(t0), ∂ju(t) ⇀ ∂ju(t0) in HN−1per (P ), t → t0.
We consider the evolution equation
∂1∂2u = f − A1(∂1 − ∂2)u =: f .
The right–hand side belongs to L∞(
[0, T ],HN−1per (P )
)
∩ C(
[0, T ],HN−2per (P )
)
, and
the ”solution” ∂2u of ∂1v = f belongs to L∞(
[0, T ],HN−1per (P )
)
, hence (∂2u)t ∈L∞(
[0, T ],HN−2per (P )
)
. Proposition 1.1 now shows that
EN−1(∂2u)(t) ≤EN−1(∂2u)(t0)eCN−1(t−t0)
+
∫ t
t0
eCN−1(t−τ)EN−1(f − A1(∂1 − ∂2)u)(τ)dτ.
This gives
lim supt→t0+0
EN−1(∂2u)(t) ≤ EN−1(∂2u)(t0) ≤ lim inft→t0
EN−1(∂2u)(t),
and, in consequence, limt→t0+0 ‖∂2u(t)‖HN−1(P ) = ‖∂2u(t0)‖HN−1(P ). It follows
that ∂2u(t) is HN−1per (P )–continuous from the right. Changing the time direc-
tion gives continuity from the left. By the proof of Lemma 3.1 we have ut ∈C(
[0, T ],HN+1per (P )
)
. �
The following proposition sharpens this result.
Proposition 2.1 Let the assumptions of the previous lemma be satisfied. Addition-ally, we suppose
σ ∈ HN+1per (P ), (2.6)
σa, σb ∈ C(
[0, T ],HN+1per (P )
)
. (2.7)
Then the solution u of (1.4), (1.5) satisfies
u ∈ C1(
[0, T ],HNper(P )
)
∩ C2(
[0, T ],HN−1per (P )
)
,
∂ju ∈ C(
[0, T ],HNper(P )
)
.
12 3 QUASILINEAR WEAKLY HYP. EQQ. WITH SPATIAL DEG.
Proof Let hε be a Friedrichs Mollifier with support [−ε, ε] and define aε(x, t) :=(a(., t) ∗ hε(.))(x), bε(x, t) := (b(., t) ∗ hε(.))(x). Then we have
aε, bε
C1([0,T ],HNper(P ))−−−−−−−−−−−→ a, b
and‖σaε‖C([0,T ],HN+1
per (P )) , ‖σbε‖C([0,T ],HN+1per (P )) ≤ C ∀ε > 0,
see Lemma A.2. It follows that∥
∥lεj∥
∥
C([0,T ],HN+1per (P )) ≤ C,
∥
∥Aεj
∥
∥
C([0,T ],HNper(P )) ≤ C
for all ε > 0. We set uε0 := u0 ∗ hε, uε
1 := u1 ∗ hε, f ε := f ∗ hε and consider theCauchy problem
utt + σbεuxt − σ2aεuxx = f ε,
u(x, 0) = uε0(x), ut(x, 0) = uε
1(x).
From the previous lemma we know that there exists a solution uε ∈C1(
[0, T ],HNper(P )
)
with ∂εju
ε ∈ C(
[0, T ],HNper(P )
)
. We have
EεN (uε)(t) ≤ C ∀ε > 0.
The same arguments as in the previous lemma give strong convergence of u, ∂ju inHN−1
per (P ), weak convergence in HNper(P ) and regularity of the limit. �
Remark 2.1 These results can be generalized to systems, see Remark 1.1.
3 Quasilinear Weakly Hyperbolic Equations with Spa-
tial Degeneracy
We study the Cauchy problem (1.5),
utt + σ(x)b(x, t, u)uxt − σ2(x)a(x, t, u)uxx = f(x, t, u, ut, σ(x)ux) (3.1)
under the assumptions N ≥ 3, (1.6), (2.6), (2.1) and
a, b,∈ C1(
[0, T ], CN+1(Kδ))
, (3.2)
f ∈ C(
[0, T ], CN (K ′δ))
, (3.3)
a, b, f are P–periodic with respect to x, (3.4)
with
Kδ :={
(x, v) ∈ R2 : x ∈ P, |v − u0(x)| ≤ δ
}
,
K ′δ :=
{
(x, v1, v2, v3) ∈ R4 : (x, v1) ∈ Kδ, |v2 − u1(x)| ≤ δ,
|v3 − σ(x)u0,x(x)| ≤ δ}
.
13
To simplify notation we define
S(T ∗) :={
v ∈ C1(
[0, T ∗],HNper(P )
)
: σvx ∈ C(
[0, T ∗],HNper(P )
)
,
(x, v(x, t), vt(x, t), σ(x)vx(x, t)) ∈ K ′δ ∀(x, t) ∈ P × [0, T ∗],
v fulfils the initial conditions (1.5)}
.
The aim of this section is to prove the
Theorem 3.1 We assume (1.6), (2.6), (2.1), (3.2)– (3.4) and N ≥ 3.
Then there exists a T ∗, 0 < T ∗ ≤ T , such that (3.1), (1.5) has a solution u ∈ S(T ∗).This solution is unique in the set of P–periodic functions.
The uniqueness follows from Hadamard’s Formula and the energy estimate (1.12).
We consider the linearized problem
L(v)u = f(v),
L(v) = ∂tt + σb(x, t, v)∂xt − σ2a(x, t, v)∂xx,
f(v) = f(x, t, v(x, t), vt(x, t), σ(x)vx(x, t))
with initial data u0, u1 and study the mapping v 7→ u.
The following, rather technical, lemma provides the equivalence between somenorms.
Lemma 3.1 Write
SN (v)(t) := ‖v(t)‖HN (P ) + ‖vt(t)‖HN (P ) + ‖σvx(t)‖HN (P ) .
Let h ∈ S(T ) and EN (h)(t) ≤ D for all t. Then there exists a constant CS,N =CS,N (D) such that
1
CS,NSN (v)(t) ≤ E (h)
N (v)(t) ≤ CS,NSN (v)(t) ∀v ∈ S(T ).
Here we used the notation
E (h)N (v)(t) = EN (v)(t) + EN (∂
(h)1 v)(t) + EN (∂
(h)2 v)(t),
∂(h)j (v) = (∂t − lj(x, t, h(x, t))∂x) v.
A proof can be found in the appendix.
Proposition 3.1 There exist constants C∗, 0 < T ∗ ≤ T , such that:
If v ∈ S(T ∗) and E (v)N (v) ≤ C∗, then there exists a solution of (3.1), (1.5) with
u ∈ S(T ∗) and E (u)N (u) ≤ C∗.
14 3 QUASILINEAR WEAKLY HYP. EQQ. WITH SPATIAL DEG.
Proof We set C∗ = 2E (u0+tu1)N (u0 + tu1)(t = 0). From v ∈ S(T ∗) we get
a(x, t, v(x, t)), b(x, t, v(x, t)) ∈ C1(
[0, T ∗],HNper(P )
)
,
σ(x)d
dxa(x, t, v(x, t)), σ(x)
d
dxb(x, t, v(x, t)) ∈ C
(
[0, T ∗],HNper(P )
)
,
f(v) ∈ C(
[0, T ∗],HNper(P )
)
and∥
∥
∥A(v)j (x, t)
∥
∥
∥
HN (P )≤ CA,N (‖v‖∞ , ‖vt‖∞ , ‖σvx‖∞)(SN (v)(t) + 1),
∥
∥
∥l(v)j (x, t)
∥
∥
∥
HN (P )≤ Cl,N (‖v‖∞ , ‖vt‖∞ , ‖σvx‖∞)(SN (v)(t) + 1),
∥
∥f(v)(x, t)∥
∥
HN (P )≤ Cf,N (‖v‖∞ , ‖vt‖∞ , ‖σvx‖∞)(SN (v)(t) + 1),
see Lemma A.1. The norms ‖v‖∞, ‖vt‖∞, ‖σvx‖∞ are bounded, since v ∈ S(T ∗).Therefore we may assume that CA,N , Cl,N , Cf,N are constants depending on δ and‖u0‖∞, ‖u1‖∞, ‖σu0,x‖∞.
Proposition 2.1 guarantees a solution u ∈ C1(
[0, T ∗],HNper(P )
)
with ∂ju ∈C(
[0, T ∗],HNper(P )
)
.
We next show that E (v)N (u)(t) ≤ 2
3C∗ if t ≤ T ∗ and T ∗ is sufficiently small. We have
E (v)N (u)(t) ≤ eDN t
(
E (v)N (u)(0)eCN t + 2
∫ t
0eCN (t−τ)EN (f(v))(τ) dτ
)
,
where DN , CN depend only on CA,N , Cl,N and C∗. We deduce that
E (v)N (u)(t) ≤ e(DN +CN )t
(
1
2C∗ + 2
∫ t
0Cf,N (CSC∗ + 1) dτ
)
≤ 2
3C∗
if T ∗ is sufficiently small. From E (v)3 (u) ≤ 2
3C∗ we conclude that
‖ut‖∞ + ‖utx‖∞ + ‖uxx‖∞ ≤ C(‖ut‖H2(P ) + ‖u‖H3(P )) ≤2
3C ′C∗,
hence
‖utt‖∞ ≤(
‖σ‖∞ ‖b‖∞ + ‖σ‖2∞ ‖a‖∞
) 2
3C ′C∗ +
∥
∥f(v)
∥
∥
∞≤ C.
The result is
|u(x, t) − u0(x)| ≤ tC, |ut(x, t) − u1(x)| ≤ tC,
|σ(x)ux(x, t) − σ(x)u0,x(x)| ≤ tC.
It follows that u ∈ S(T ∗) if T ∗ is sufficiently small.
It remains to show that E (u)N (u) ≤ C∗. Therefore we prove that
‖(lj(x, t, v(x, t)) − lj(x, t, u(x, t)))ux‖HN (P ) ≤1
6C∗. (3.5)
15
We denote the left–hand side of (3.5) by cj(x, t) and have
cj(x, t) ≤ Cprod,N ‖αj(x, t)‖HN (P ) ‖σux‖HN (P )
≤ 2
3Cprod,NCSC∗ ‖αj(x, t)‖
HN (P ) ,
where αj(x, t) = βj(x, t, v(x, t)) − βj(x, t, u(x, t)). From dt ‖α(., t)‖HN (P ) =‖αt(., t)‖HN (P ), α(x, 0) = 0,
αt(x, t) =βj,t(x, t, v(x, t)) − βj,t(x, t, u(x, t))
+ βj,v(x, t, v(x, t))vt(x, t) − βj,v(x, t, u(x, t))ut(x, t)
and ‖vt‖HN (P ), ‖ut‖HN (P ) ≤ CSC∗ we obtain
‖α(., t)‖HN (P ) ≤∫ t
0‖ατ (., τ)‖HN (P ) dτ ≤ Ct.
Thus we have (3.5). �
From this lemma it may be concluded that there exists a sequence (un) ⊂ S(T ∗)with
L(un−1)un = f(un−1).
Lemma 3.2 The sequences (un), (unt ), (σun
x) are Cauchy sequences in the spaceC(
[0, T ∗],HN−1per (P )
)
if T ∗ is sufficiently small.
Proof It holds
L(un−1)(un − un+1) = f(un−1) − f(un) +(
L(un) − L(un−1))
un+1.
Using Hadamard’s Formula one can estimate the L2(P )–norm of the right–hand sideby CS0(u
n − un−1)(t). It follows that
E (un−1)0 (un − un+1)(t) ≤ C ′
∫ t
0S0(u
n − un−1)(τ) dτ.
Hence we obtain
maxt∈[0,T ∗]
S0(un − un+1)(t) ≤ C ′′T ∗ max
t∈[0,T ∗]S0(u
n − un−1)(t).
We apply the Interpolation Theorem of Gagliardo–Nirenberg and the proof is com-plete. �
Consequently, there exists a limit u ∈ C(
[0, T ∗],HN−1per (P )
)
with ∂ju ∈C(
[0, T ∗],HN−1per (P )
)
. It is immediate that u is a solution. The boundedness in
HNper(P ) implies
un(., t) ⇀ u(., t), unt (., t) ⇀ ut(., t), σun
x(., t) ⇀ σux(., t)
16 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
in HNper(P ). This clearly forces u, ut, ∂ju ∈ L∞
(
[0, T ∗],HNper(P )
)
, even u ∈C(
[0, T ∗],HNper(P )
)
. It remains to prove that ∂ju ∈ C(
[0, T ∗],HNper(P )
)
. We have
∂(u)1 ∂
(u)2 u = f(u) − A
(u)1
(
∂(u)1 − ∂
(u)2
)
u =: f .
The right–hand side belongs to L∞(
[0, T ∗],HNper(P )
)
∩ C(
[0, T ∗],HN−1per (P )
)
, the
”solution” ∂(u)2 u of ∂
(u)1 v = f belongs to L∞
(
[0, T ∗],HNper(P )
)
and we have l(u)1 (x, t) ∈
C(
[0, T ∗],HNper(P )
)
. Hence we can apply Proposition 1.1 and obtain
EN (∂(u)2 u)(t) ≤EN (∂
(u)2 u)(t0)e
CN (t−t0)
+
∫ t
t0
eCN (t−τ)EN
(
f(u) − A(u)1
(
∂(u)1 − ∂
(u)2
)
u)
(τ) dτ.
This gives
lim supt→t0+0
EN (∂(u)2 u)(t) ≤ EN (∂u
2 u)(t0) ≤ lim inft→t0
EN (∂(u)2 u)(t),
which implies the HNper(P )–continuity from the right of ∂
(u)2 u. Changing the time
direction completes the proof of Theorem 3.1.
Remark 3.1 One can prove a similar result for the system
ui,tt + σ(x)b(x, t, ~u)ui,xt − σ2(x)a(x, t, ~u)ui,xx
= f(x, t, ~u, ~ut, σ(x)~ux), i = 1, . . . , n
~u(x, 0) = ~u0(x), ~ut(x, 0) = ~u1(x),
if one uses the energy
E (~v)N (~u) :=
n∑
i=1
E (~v)N (ui).
4 Quasilinear Weakly Hyperbolic Equations with
Time– and Spatial Degeneracy
In this section we will derive an existence result for the quasilinear Cauchy problemwith spatial– and time–degeneracy (1.5),
utt(x, t) + λ(t)σ(x)b(x, t, u(x, t))uxt(x, t) (4.1)
− λ(t)2σ(x)2a(x, t, u(x, t))uxx(x, t)
= f(x, t, u(x, t), ut(x, t), λ′(t)σ(x)ux(x, t)).
Let λ(t) ∈ C1([0, T ]) be a function satisfying (0.4).
The aim of this section is to prove the
4.1 The Reduction Process 17
Theorem 4.1 We suppose (2.1) and
b2(x, t, u) + 4a(x, t, u) ≥ γ > 0 (x, t, u) ∈ Kδ, (4.2)
σ ∈ HN+2per (P ), (4.3)
a, b ∈ C1(
[0, T ], C∞(Ru) × HNper(P )
)
, (4.4)
σa, σb ∈ C(
[0, T ], C∞(Ru) × HN+1per (P )
)
, (4.5)
f ∈ C(
[0, T ], C∞(R3u,ut,λ′σux
) × HNper(P )
)
. (4.6)
Then there exist constants T ∗ > 0 and r ∈ N, such that: If
N ≥ r + 3, (4.7)
then there exists a solution u with u, ut, σux ∈ C(
[0, T ∗],HN−rper (P )
)
. The number rdescribes the loss of Sobolev regularity and may depend on N . If N is sufficientlylarge, then (4.7) holds.
We divide the proof into three steps. At first we present a special technique totransform the quasilinear problem to another problem whose right–hand side hasa suitable asymptotic. Then we consider linear weakly hyperbolic problems withspecial right–hand side and show an existence result and an a priori estimate. Afterthat we study a linearized version of the new quasilinear problem and construct amapping of functions. Using the results of the second step we construct a sequenceof such functions and prove the convergence to a solution.
4.1 The Reduction Process
Let u be a solution of (4.1), (1.5). We study the system of ordinary differentialequations with parameter x
u(0)tt (x, t) = f
(
x, t, u(0)(x, t), u(0)t (x, t), 0
)
,
u(i)tt (x, t) = gi
(
x, t, u(i)(x, t), u(i)t (x, t)
)
:=
18 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
f
x, t,
i∑
j=0
u(j)(x, t),
i∑
j=0
u(j)t (x, t), σ(x)λ′(t)
i−1∑
j=0
u(j)x (x, t)
− f
x, t,i−1∑
j=0
u(j)(x, t),i−1∑
j=0
u(j)t (x, t), σ(x)λ′(t)
i−2∑
j=0
u(j)x (x, t)
− λ(t)σ(x)bi−1(x, t)i−1∑
j=0
u(j)xt (x, t) + λ(t)σ(x)bi−2(x, t)
i−2∑
j=0
u(j)xt (x, t)
+ λ(t)2σ(x)2ai−1(x, t)
i−1∑
j=0
u(j)xx (x, t) − λ(t)σ(x)ai−2(x, t)
i−2∑
j=0
u(j)xx (x, t),
i = 1, . . . , p, N − 2p ≥ 3,
−1∑
j=0
= 0.
with the initial data
u(0)(x, 0) = u0(x), u(0)t (x, 0) = u1(x),
u(i)(x, 0) = u(i)t (x, 0) = 0.
Here we used the notation
bk(x, t) = b
x, t,
k∑
j=0
u(j)(x, t)
, ak(x, t) = a
x, t,
k∑
j=0
u(j)(x, t)
.
This system may be interpreted as a system of weakly hyperbolic equations withspatial degeneracy, whose degenerating function vanishes identically. Theorem 3.1implies, that these equations have solutions u(i) ∈ C2
(
[0, Ti],HN−2iper (P )
)
with
∣
∣
∣u(0)(x, t) − u0(x)
∣
∣
∣+∣
∣
∣u
(0)t (x, t) − u1(x)
∣
∣
∣
+∣
∣
∣σ(x)u(0)
x (x, t) − σ(x)u0,x(x)∣
∣
∣≤ ε0,
∣
∣
∣u(i)(x, t)∣
∣
∣+∣
∣
∣u(i)t (x, t)
∣
∣
∣+∣
∣
∣σ(x)u(i)x (x, t)
∣
∣
∣ ≤ εi (i ≥ 1).
Furthermore, there are constants Ci with∥
∥u(i)∥
∥
C2([0,Ti],HN−2iper (P )) ≤ Ci.
We want to show the following
Lemma 4.1 It holds
FN−2i
(
u(i))
(t) ≤ Ciλ(t)i.
Proof The assertion is true for i = 0.
4.1 The Reduction Process 19
Let FN−2(i−1)
(
u(i−1))
(t) ≤ Ci−1λ(t)i−1. We want to apply Lemma 1.1 and for thispurpose we study the right–hand side gi: From Hadamard’s Formula we see that
gi(x, t, u(i), u(i)t ) =f1i(x, t)u(i) + f2i(x, t)u
(i)t + σ(x)λ′(t)f3i(x, t)u(i−1)
x
+ σ(x)λ(t)b1i(x, t)u(i−1)xt + σ(x)λ(t)b2i(x, t)u(i−1)
+ σ(x)2λ(t)2a1i(x, t)u(i−1)xx + σ(x)2λ(t)2a2i(x, t)u(i−1).
Hence we obtain
FN−2i
(
u(i))
(t) ≤ C
∫ t
0λ′(τ)
∥
∥
∥u(i−1)(τ)
∥
∥
∥
HN+1−2i(P )+ λ(τ)
∥
∥
∥u(i−1)
τ (τ)∥
∥
∥
HN+1−2i(P )
+ λ(τ)∥
∥
∥u(i−1)(τ)∥
∥
∥
HN−2i(P )+ λ(τ)2
∥
∥
∥u(i−1)(τ)∥
∥
∥
HN+2−2i(P )dτ
≤ C
∫ t
0λ′(τ)λ(τ)i−1 + λ(τ)λ(τ)i−1 + λ(τ)2λ(τ)i−1 dτ
≤ Ciλ(t)i. �
We define u =:∑p
j=0 u(j) + v and
bk(x, t, v) = b
x, t,
k∑
j=0
u(j) + v
, ak(x, t, v) = a
x, t,
k∑
j=0
u(j) + v
.
From (4.1) we see that
vtt + λ(t)σ(x)bp(x, t, v)vxt − λ(t)2σ(x)2ap(x, t, v)vxx
= f
x, t,
p∑
j=0
u(j) + v,
p∑
j=0
u(j)t + vt, λ
′(t)σ(x)
p∑
j=0
u(j)x + vx
− f
x, t,
p∑
j=0
u(j),
p∑
j=0
u(j)t , λ′(t)σ(x)
p−1∑
j=0
u(j)x
− λ(t)σ(x)bp(x, t, v)
p∑
j=0
u(j)xt + λ(t)σ(x)bp−1(x, t, 0)
p−1∑
j=0
u(j)xt
+ λ(t)2σ(x)2ap(x, t, v)
p∑
j=0
u(j)xx − λ(t)2σ(x)2ap−1(x, t, 0)
p−1∑
j=0
u(j)xx
=: Fp(x, t, v, vt, λ′(t)σ(x)vx).
The purpose for these considerations is the following
Proposition 4.1 It holds
‖Fp(x, t, 0, 0, 0)‖HN−2−2p(P ) ≤ Cλ(t)p.
20 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
Proof We write Fp(x, t, 0, 0, 0) = D1 + D2 + D3, with
D1 := f(x, t,
p∑
,
p∑
,
p∑
) − f(x, t,
p∑
,
p∑
,
p−1∑
),
D2 := −λσbp(x, t, 0)(
p∑
. . . )xt + λσbp−1(x, t, 0)(
p−1∑
. . . )xt
D3 := λ2σ2ap(x, t, 0)(
p∑
. . . )xx − λ2σ2ap−1(x, t, 0)(
p−1∑
. . . )xx.
From Hadamard’s Formula, Lemma 4.1 and
D2 = −λσbp(x, t, 0)u(p)xt − λσ(bp(x, t, 0) − bp−1(x, t, 0))(
p−1∑
. . . )xt,
D3 = λ2σ2ap(x, t, 0)u(p)xx + λ2σ2(ap(x, t, 0) − ap−1(x, t, 0))(
(p−1)∑
. . . )xx
one has the assertion. �
4.2 Linear Theory for equations with special right–hand side
Now we are able to derive an existence result and an a priori estimate for linearequations with spatial– and time–degeneracy.
Proposition 4.2 Let λ(t) ∈ C1([0, T ]) be a function satisfying (0.4). We assumeN ≥ 3, (1.6), (4.3), (1.10), (2.7)
f
λd−1λ′∈ C
(
[0, T ],HNper(P )
)
,
d > Q := sup(x,t)∈P×[0,T ]
∣
∣
∣
∣
∣
b(x, t)√
b2(x, t) + 4a(x, t)
∣
∣
∣
∣
∣
+ 1.
Then there exists a solution u of
utt + σ(x)λ(t)b(x, t)uxt − σ2(x)λ2(t)a(x, t)uxx = f(x, t), (4.8)
u(x, 0) = ut(x, 0) = 0 (4.9)
with u ∈ C1(
[0, T ],HNper(P )
)
and λ−du, σux ∈ C(
[0, T ],HNper(P )
)
. The followingestimates hold:
EM (u)′(t) ≤(
CM + Q′λ′(t)
λ(t)
)
EM (u)(t) + 2EM (f)(t), M = 0, . . . , N − 1,
EN (u)(t) ≤ 2
∫ t
0eCN (t−τ)
(
λ(t)
λ(τ)
)Q′
EN (f)(τ)dτ,
4.2 Linear Theory for equations with special right–hand side 21
where Q < Q′ < d. The constants CM depend on
‖lj‖C
“
[0,T ],Hmax(2,M)per (P )
” ,
‖a‖C1
“
[0,T ],Hmax(1,M)per (P )
” , ‖b‖C1
“
[0,T ],Hmax(1,M)per (P )
” ,
‖σax‖C
“
[0,T ],Hmax(1,M)per (P )
” , ‖σbx‖C
“
[0,T ],Hmax(1,M)per (P )
” .
Proof We approximate
C1(
[0, T ],HN+2per (P )
)
3 aε, bε
C1([0,T ],HNper(P ))−−−−−−−−−−−→ a, b,
C(
[0, T ],HN+1per (P )
)
3 fε
C([0,T ],HNper(P ))−−−−−−−−−−−→ f
λd−1λ′,
such that
‖σaε,x‖HN (P ) , ‖σbε,x‖HN (P ) ≤ C, b2ε + 4aε ≥ γ
2∀ 0 < ε ≤ ε0,
see Lemma A.2. We set bε := bε(λ + ε), aε := aε(λ + ε)2 and obtain b2ε + 4aε ≥
12ε2γ > 0. Due to Proposition 2.1 there exists a solution uε of
uεtt + σbεu
εxt − σ2aεu
εxx = λd−1λ′fε,
uε(x, 0) = uεt (x, 0) = 0
with uε, uεt , σuε
x ∈ C(
[0, T ],HN+1per (P )
)
. It follows that
∂εj = ∂t − lεj(x, t)∂x,
lε1,2(x, t) =1
2σ(x)(λ(t) + ε)
(
−bε(x, t) ±√
b2ε(x, t) + 4aε(x, t)
)
,
(∂ε1∂
ε2 + Aε
1 (∂ε1 − ∂ε
2))uε = λd−1λ′fε,
Aε1(x, t) = c1,ε(x, t) +
λ′(t)
λ(t) + εc2,ε(x, t),
c1,ε =−bε + Sε
4Sε(σ(−bε − Sε))x(λ(t) + ε) +
(bε + Sε)t2Sε
,
c2,ε =bε + Sε
2Sε,
with Sε =√
b2ε + 4aε. We have
lεj , c2,ε ∈ C1(
[0, T ],HN+2per (P )
)
, c1,ε ∈ C(
[0, T ],HN+1per (P )
)
and
∥
∥lεj∥
∥
C1([0,T ],HNper(P ))
, ‖c1,ε‖C([0,T ],HNper(P )) , ‖c2,ε‖C1([0,T ],HN
per(P )) ≤ C ∀ε.
22 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
Now we can derive estimates for uε. Obviously,
∂ε1∂
ε2u
ε = λd−1λ′fε − Aε1 (∂ε
1 − ∂ε2) uε,
where the right–hand side belongs to C(
[0, T ],HN+1per (P )
)
and the solution ∂ε2u
ε
belongs to C(
[0, T ],HN+1per (P )
)
. From Proposition 1.1 we get for 0 ≤ M ≤ N
EM (∂ε2u
ε)′(t) ≤ CMEM (∂ε2u
ε)(t) + EM
(
λd−1λ′fε − Aε1 (∂ε
1 − ∂ε2) uε
)
(t).
We estimate the second energy on the right:∥
∥∂Mx (Aε
1 (∂ε1 − ∂ε
2)uε)∥
∥
2
≤∥
∥∂Mx (c1,ε (∂ε
1 − ∂ε2) uε)
∥
∥
2+
λ′(t)
λ(t) + ε
∥
∥c2,ε∂Mx (∂ε
1 − ∂ε2)uε
∥
∥
2
+ C
M−1∑
j=0
λ′(t)
λ(t) + ε
∥
∥
(
∂M−jx c2,ε
)
∂jx (∂ε
1 − ∂ε2) uε
∥
∥
2
≤ CEεM (uε)(t) +
λ′(t)
λ(t) + ε
Q + ε′
2
(∥
∥∂Mx (∂ε
1uε)∥
∥
2+∥
∥∂Mx (∂ε
2uε)∥
∥
2
)
+ Cλ′(t)
λ(t) + ε‖(∂ε
1 − ∂ε2)uε‖HM−1(P )
≤ CEεM (uε)(t) +
λ′(t)
λ(t) + ε
Q + ε′
2
(∥
∥∂Mx (∂ε
1uε)∥
∥
2+∥
∥∂Mx (∂ε
2uε)∥
∥
2
)
.
The result is
EM (∂ε2u
ε)′(t) ≤CMEεM (∂ε
2uε)(t) + λd−1(t)λ′(t)EN
(
fε
)
(t)
+Q + ε′
2
λ′(t)
λ(t) + εEε
M (uε)(t).
We can prove a similar inequality for EM (∂ε1u
ε)′(t), if we use
∂ε2∂
ε1u
ε = λd−1λ′fε − Aε2 (∂ε
1 − ∂ε2) uε.
We have to replace c2,ε = bε+Sε
2Sεby bε−Sε
2Sε. By standard technique one shows an
estimate for EM (u)′(t). Combining these inequalities we obtain
EεM (uε)′(t)
≤ CMEεM (uε)(t) + 2λd−1(t)λ′(t)EM (fε)(t) + (Q + ε′)
λ′(t)
λ(t) + εEε
M (uε)(t).
We may assume that ε0 is so small that Q + ε′ ≤ Q′ < d for 0 < ε ≤ ε0. We wantto apply the Lemma of Nersesjan. Before we can do this, we have to check whetherEε
M (uε)(t) ≤ Cελ(t)d. Therefore, we apply Gronwall’s Lemma to
EεM (uε)′(t) ≤ C
εEε
M (uε)(t) + Cλd−1(t)λ′(t)
4.2 Linear Theory for equations with special right–hand side 23
and get
EεM (uε)(t) ≤ C
ε
∫ t
0e
Cε
(t−τ)λd−1(τ)λ′(τ)dτ ≤ Cελ(t)d. (4.10)
So we can apply Nersesjan’s Lemma with
y(t) := EεM (uε)(t), K(t) := Q′λ
′(t)
λ(t)+ CM , f := 2λd−1(t)λ′(t)EM (fε)(t)
and deduce that
EεM (uε)(t) (4.11)
≤ 2
∫ t
0
(
λ(t)
λ(τ)
)Q′
eCM (t−τ)λd−1(τ)λ′(τ)EM (fε)(τ)dτ ≤ Cλ(t)d.
This inequality holds for all ε in contradistinction to (4.10). We conclude
‖uε(t)‖HN (P ) + ‖uεt (t)‖HN (P ) + ‖(λ(t) + ε)σuε
x(t)‖HN (P ) ≤ Cλ(t)d,
hence
‖uε(t)‖HN (P ) + ‖uεt (t)‖HN (P ) +
∥
∥(λ(t) + ε′)σuεx(t)
∥
∥
HN (P )≤ Cλ(t)d
for 0 ≤ ε′ < ε, which implies Eε′
N (uε)(t) ≤ Cλ(t)d.
We next claim that the sequence (uε) converges in suitable Sobolev spaces. Bydefinition of uε,
Lεuε = λd−1λ′fε,
Lε′(
uε′ − uε)
= λd−1λ′(
fε′ − fε
)
+(
Lε − Lε′)
uε =: gε,ε′(x, t).
From ‖uεxt‖HN−1(P ) ≤ Cλ(t)d and ‖uε
xx‖HN−2(P ) ≤ Cλ(t)d we deduce that∥
∥gε,ε′∥
∥
HN−2(P )≤ Cλ(t)d + Cλ(t)d−1λ′(t). Without loss of generality we obtain
for 0 < ε′ < ε the estimate Eε′
N−2(uε − uε′)(t) ≤ C(ε + ε′)λ(t)d.
It follows that the sequences (uε), (uεt ), (λσuε
x) converge to limits u, ut, λσux inC(
[0, T ],HN−2per (P )
)
. By the Interpolation Theorem of Nirenberg–Gagliardo we have
convergence even in C(
[0, T ],HN−1per (P )
)
. The weak convergence of this sequences
in HNper(P ) implies
u, ut, σux ∈ L∞(
[0, T ],HNper(P )
)
, EN (u)(t) ≤ Cλ(t)d.
By standard arguments one obtains for t0 > 0
lim supt→t0+0
‖(∂ju)(t)‖HN (P ) ≤ ‖(∂ju)(t0)‖HN (P ) ≤ lim inf
t→t0‖(∂ju)(t)‖
HN (P ) ,
hence HN (P )–continuity from the right. The continuity from the left is proved bythe same technique after changing the time–direction.
24 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
The continuity for t0 = 0 is trivial, since ‖u(t)‖HN (P )+‖ut(t)‖HN (P )+‖σux(t)‖HN (P )
is obviously continuous for t0 = 0.
Finally, from (4.11) it follows that
EN (u)(t) ≤ lim infε→0
EεN (uε)(t) ≤ 2
∫ t
0
(
λ(t)
λ(τ)
)Q′
eCN (t−τ)EN (f)(τ)dτ. �
4.3 Quasilinear Equations
Proof of Theorem 4.1 First, we have to compute the number of required reductionsteps. Set
Q := sup(x,t,u)∈Kδ×[0,T ]
∣
∣
∣
∣
∣
b(x, t, u)√
b(x, t, u)2 + 4a(x, t, u)
∣
∣
∣
∣
∣
+ 1
and choose Q′ > Q. We define
Q1 := sup(x,t,u1,u2,u3)∈K′
δ×[0,T ]
|∂5f(x, t, u1, u2, u3)| ,
where ∂5 denotes the derivative with respect to the 5th argument. It is worthpointing out that
∂5Fp(x, t, v, vt, σλ′vx) = ∂5f(x, t,
p∑
u(j) + v,
p∑
u(j)t + vt, σλ′(
p∑
u(j)x + vx)).
Now we choose p ∈ N, p > Q′ with
2Q1CS,N−2−2p
p − Q′≤ 1
4. (4.12)
It may happen that for given N such a number p does not exist. But it is possibleto find p if N is large. Namely, we fix q ∈ 2N, q ≥ 4. For even N we definep := 1
2(N − 2 − q). The condition (4.12) is fulfilled, if N (and hence p) is large.The number p is the number of steps in the reduction process. We set r := 2p + 3,M := N − r + 1 ≥ 4, where N and r are the constants from Theorem 4.1.
We choose δ0 > 0 such that, if SK(v) = ‖v‖HK(P ) + ‖vt‖HK(P ) + ‖σλvx‖HK(P ) ≤ δ0
and K ≥ 1, then ‖v‖∞ ≤ δ2 , ‖vt‖∞ ≤ δ
2 , ‖σλvx‖∞ ≤ δ2 .
Let Tp > 0 be a constant such that∣
∣
∣
∣
∣
p∑
u(j)(x, t) − u0(x)
∣
∣
∣
∣
∣
,
∣
∣
∣
∣
∣
p∑
u(j)t (x, t) − u1(x)
∣
∣
∣
∣
∣
,
∣
∣
∣
∣
∣
σλ′
(
p∑
u(j)x (x, t) − u0,x(x)
)∣
∣
∣
∣
∣
≤ δ
2, t ∈ [0, Tp].
We study the reduced problem (4.9),
L(v)p u = utt + σλbp(x, t, v)uxt − σ2λ2ap(x, t, v)uxx = Fp(x, t, v, vt, σλ′vx), (4.13)
with SM (v)(t) ≤ δ0 for t ∈ [0, Tp] and v(x, 0) = vt(x, 0) = 0.
We will prove the
4.3 Quasilinear Equations 25
Lemma 4.2 There exists a constant 0 < T0 ≤ Tp with:
If SM (v)(t) ≤ λ(t)p, then SM (u)(t) ≤ λ(t)p for 0 ≤ t ≤ T0.
Proof Let CM be the constant from Proposition 4.2, such that for SM(v) ≤ δ0 and
L(v)p u = g it holds
E (v)M (u)(t) ≤ 2
∫ t
0eCM (t−τ)
(
λ(t)
λ(τ)
)Q′
EM (g)(τ)dτ.
We choose constants CF,1 and CF,2 with
‖Fp(x, t, 0, 0, 0)‖HM (P ) ≤CF,1λ(t)p,
‖Fp(x, t, v1, v2, v3)‖HM (P ) ≤CF,2(‖v1‖HM (P ) + ‖v2‖HM (P ) + ‖v3‖HM−1(P ))
+ Q1
∥
∥∂Mx v3
∥
∥
2+ CF,1λ(t)p
for ‖vj‖∞ ≤ δ2 , see Proposition 4.1 and Lemma A.1. Let T0 satisfy the following
conditions
λ(T0) ≤ δ0,
CP λ(T0)p ‖σ‖∞
∥
∥λ′∥
∥
∞≤ δ
2,
CP,M−1 ‖σ‖HM−1(P )
∥
∥λ′∥
∥
∞≤ 1,
eCM T0 ≤ 2,
2(2CF,2 + CF,1)CS,MT0 ≤ 1
4,
where CP and CP,M−1 are the imbedding constants from Section 1. Now we showthat SM (u)(t) ≤ λ(t)p for t ≤ T0.
The vector (x, t, v, vt, σλ′vx) lies in the domain of Fp, since ‖v‖HM (P ) ≤ δ0,‖vt‖HM (P ) ≤ δ0 and
∥
∥σλ′vx
∥
∥
∞≤ ‖σ‖∞
∥
∥λ′∥
∥
∞CP ‖v‖H2(P ) ≤ CP ‖σ‖∞
∥
∥λ′∥
∥
∞SM (v) ≤ δ
2.
Hence we obtain
∥
∥Fp(x, t, v, vt, σλ′vx)∥
∥
HM (P )≤ CF,2(‖v‖HM (P ) + ‖vt‖HM (P ) +
∥
∥σλ′vx
∥
∥
HM−1(P ))
+ Q1
∥
∥∂Mx (σλ′vx)
∥
∥
2+ CF,1λ(t)p.
From
∥
∥σλ′vx
∥
∥
HM−1(P )≤ CP,M−1 ‖σ‖HM−1(P )
∥
∥λ′∥
∥
∞‖v‖HM (P ) ≤ ‖v‖HM (P ) ,
∥
∥∂Mx (σλ′vx)
∥
∥
2≤ λ′(t)
λ(t)‖σλvx‖HM (P ) ≤
λ′(t)
λ(t)SM (v)(t)
26 4 QUASILIN. W. HYP. EQQ. WITH TIME – AND SPATIAL DEG.
we deduce that∥
∥Fp(x, t, v, vt, σλ′vx)∥
∥
HM (P )≤ (2CF,2 + CF,1)λ(t)p + Q1λ
′(t)λ(t)p−1.
By EM (. . . ) = ‖. . . ‖HM (P ) and the choice of p and T0 we can assert that
CS,ME (v)M (u)(t)
≤ 2CS,M
∫ t
0eCM (t−τ)
(
λ(t)
λ(τ)
)Q′
EM (Fp(x, τ, v, vt, σλ′vx))dτ
≤ CS,MeCM T0λ(t)Q′
∫ t
0λ(τ)−Q′
2((2CF,2 + CF,1)λ(τ)p + Q1λ′(τ)λ(τ)p−1)dτ
≤ CS,MeCM T0λ(t)p2(2CF,2 + CF,1)t + CS,MeCM T0λ(t)p 2Q1
p − Q′
≤ λ(t)p.
The assertion follows from SM (u) ≤ CSE (v)M (u). The Lemma is proved. �
Thus, we can define a sequence (vi) ⊂ C1(
[0, T0],HMper(P )
)
with ∂jvi ∈C(
[0, T0],HMper(P )
)
such that
v0 ≡ 0,
L(vi)p vi+1 = Fp(x, t, vi, vi,t, σλ′vi,x),
vi(x, 0) = vi,t(x, 0) = 0.
These functions satisfy SM (vi)(t) ≤ λ(t)p. We will prove a convergence result.
Lemma 4.3 The sequence (vi) is a Cauchy sequence in the Banach spaceC1(
[0, T0],HM−1per (P )
)
.
Proof We define wi = vi+1 − vi and have
wi,tt + λσb(x, t, vi−1)wi,xt − λ2σ2a(x, t, vi−1)wi,xx
= Fp(x, t, vi, vi,t, σλ′vi,x) − Fp(x, t, vi−1, vi−1,t, σλ′vi−1,x)
+ λσ(bp(x, t, vi−1) − bp(x, t, vi))vi+1,xt − λ2σ2(ap(x, t, vi−1) − ap(x, t, vi))vi+1,xx
= Fp(x, t, vi, vi,t, σλ′vi,x) − Fp(x, t, vi−1, vi−1,t, σλ′vi−1,x)
+ λσbi(x, t)wi−1vi+1,xt + λ2σ2ai(x, t)wi−1vi+1,xx
=: gi(x, t)
By Hadamard’s Formula and the choice of Q1 we have∥
∥Fp(x, t, vi, vi−1,t, σλ′vi−1,x) − Fp(x, t, vi−1, vi−1,t, σλ′vi−1,x)∥
∥
2+
+∥
∥Fp(x, t, vi, vi,t, σλ′vi−1,x) − Fp(x, t, vi, vi−1,t, σλ′vi−1,x)∥
∥
2≤ CfS0(wi−1),
∥
∥Fp(x, t, vi, vi,t, σλ′vi,x) − Fp(x, t, vi, vi,t, σλ′vi−1,x)∥
∥
2≤ Q1
λ′(t)
λ(t)S0(wi−1),
‖λσbi(x, t)wi−1vi+1,xt‖2 ≤ CPCb ‖σ‖∞ λ(t)p+1S0(wi−1),∥
∥λ2σ2ai(x, t)wi−1vi+1,xx
∥
∥
2≤ CP Ca ‖σ‖2
∞ λ(t)p+2S0(wi−1).
27
With the assumption
2eC0T0CS,0CKT0
:= 2eC0T0CS,0(Cf + CP Cb ‖σ‖∞ λ(T0)p+1 + CPCa ‖σ‖2
∞ λ(T0)p+2)T0 ≤ 1
4
we obtain
‖gi‖2 ≤ CKS0(wi−1) + Q1λ′(t)
λ(t)S0(wi−1).
We suppose S0(wi−1)(t) ≤ Cwi−1λ(t)p. Without loss of generality we may assume
that the sequences (Cq), (CS,q) are monotonically increasing with q. From this andProposition 4.2 it follows that
CS,0E0(wi)(t) ≤ 2CS,0
∫ t
0eC0(t−τ)
(
λ(t)
λ(τ)
)Q′
‖gi‖2 (τ)dτ
≤ Cwi−1
(
2eC0T0CS,0CKt +2Q1CS,0
p − Q′eC0T0
)
λ(t)p
≤ 3
4Cw
i−1λ(t)p.
Hence we obtain S0(wi−1)(t) ≤ Cw0
(
34
)iλ(t)p. Nirenberg–Gagliardo Interpolation
and SM (wi)(t) ≤ 2λ(t)p give the assertion. The Lemma is proved. �
By standard arguments one can show that the limit v is a solution of the equation
L(v)p v = vtt + σλbp(x, t, v)vxt − σ2λ2ap(x, t, v)vxx = Fp(x, t, v, vt, σλ′vx).
It follows that u =∑p u(j) + v solves (4.1) and (1.5). The Theorem is proved. �
Remark 4.1 Obviously, one can show a similar theorem for systems with diagonalprincipal part.
5 Fully Nonlinear Weakly Hyperbolic Equations
It is sufficient to show that a fully nonlinear weakly hyperbolic Cauchy problem isequivalent to a suitably chosen weakly hyperbolic quasilinear Cauchy system, seeRemarks 1.1, 2.1, 3.1, 4.1. We will divide this proof into two parts.
Theorem 5.1 Let
Fj ∈ C1([0, T ], C∞(R6) × H5per(P )),
u, ut, σλux ∈ C2(
[0, T ],H5per(P )
)
,
λ ∈ C2([0, T ]), σ ∈ H7per(P ).
28 5 FULLY NONLINEAR WEAKLY HYPERBOLIC EQUATIONS
The function u is a solution of the Cauchy problem (0.1), (0.2), if and only if(u0, u1, u2) := (u, ut, ux) is a solution of the system
F1u1,tt + σλF2u1,xt + σ2λ2F3u1,xx + σλ′F4u1,x + F5u1,t + F6u1 + F8
+ σλ′F2u2,t + σ2(λ2)′F3u2,x + σλ′′F4u2 = 0, (5.1)
F1u2,tt + σλF2u2,xt + σ2λ2F3u2,xx + σλ′F4u2,x + F5u2,t + F6u2 + F7
+ σ′λF2u2,t + (σ2)′λ2F3u2,x + σ′λ′F4u2 = 0, (5.2)
F1u0,tt + σλF2u0,xt + σ2λ2F3u0,xx + σλ′F4u0,x
−F1u1,t − σλF2u2,t − σ2λ2F3u2,x − σλ′F4u2 + F = 0, (5.3)
u1(x, 0) = ϕ1(x), u1,t(x, 0) = ϕ2(x),
u2(x, 0) = ϕ0,x(x), u1,t(x, 0) = ϕ1,x(x),
u0(x, 0) = ϕ0(x), u1,t(x, 0) = ϕ1(x),
where the function F and the derivatives Fj depend on
(u1,t, σλu2,t, σ2λ2u2,x, σλ′u2, u0,t, u0, x, t).
For the definition of ϕ2 see the assumption A 8.
Proof We restrict us to show the ⇐–direction.
Differentiating (5.3) with respect to t and subtraction from (5.1) gives
F1(u1 − u0,t)tt + F2σλ(u1 − u0,t)xt + F3σ2λ2(u1 − u0,t)xx + F4σλ′(u1 − u0,t)x
− (F1,t(u0,t − u1)t + F2,tσλ(u0,x − u2)t + F3,tσ2λ2(u0,x − u2)x)
− (F4,tσλ′(u0,x − u2) + F2σλ′(u0,x − u2)t + F3σ2(λ2)′(u0,x − u2)x)
− F4σλ′′(u0,x − u2) + F5(u1 − u0,t)t + F6(u1 − u0,t) = 0. (5.4)
Differentiating (5.3) with respect to x and subtracting from (5.2) yields
F1(u2 − u0,x)tt + F2σλ(u2 − u0,x)xt + F3σ2λ2(u2 − u0,x)xx + F4σλ′(u2 − u0,x)x
− (F1,x(u0,t − u1)t + F2,xσλ(u0,x − u2)t + F3,xσ2λ2(u0,x − u2)x)
− (F4,xσλ′(u0,x − u2) + F2σ′λ(u0,x − u2)t + F3(σ
2)′λ2(u0,x − u2)x)
− F4σ′λ′(u0,x − u2) + F5(u2 − u0,x)t + F6(u2 − u0,x) = 0. (5.5)
The system (5.4), (5.5) is a linear homogeneous weakly hyperbolic system for thefunctions v1 = u0,t − u1, v2 = u0,x − u2. The Levi conditions are satisfied. Fromthe uniqueness of periodic solutions we get v1 ≡ v2 ≡ 0. Combining this resultwith (5.3) gives the assertion. �
The next theorem will reduce the quasilinear principal part to a semilinear one.
29
Theorem 5.2 We assume
a, b ∈ C1([0, T ], C∞(Ru) × H4per(P )),
f ∈ C([0, T ], C∞(Ru) × H4per(P )),
~u, ~ut, σλ~ux ∈ C1(
[0, T ],H4per(P )
)
,
λ ∈ C2([0, T ]).
The vector ~u is a solution of
~utt + σλb(~u, ~ut, σλ′~ux, x, t)~uxt − σ2λ2a(~u, ~ut, σλ′~ux, x, t)~uxx
= ~f(~u, ~ut, σλ′~ux, x, t),
~u(x, 0) = ~ϕ0(x), ~ut(x, 0) = ~ϕ1(x)
if and only if the vector (~u0, ~u1, ~u2) := (~u, ~ut, ~ux) is a solution of
~u1,tt + σλb~u1,xt − σ2λ2a~u1,xx + σλbt~u2,t + σλ′b~u2,t (5.6)
− σ2λ2at~u2,x − σ2(λ2)′a~u2,x = ~ft,
~u2,tt + σλb~u2,xt − σ2λ2a~u2,xx + σλbx~u2,t + σ′λb~u2,t (5.7)
− σ2λ2ax~u2,x − (σ2)′λ2a~u2,x = ~f1~u2 + ~f2~u2,t + ~f3(σλ′~u2)x + ~f4,
bx = b1~u2 + b2~u2,t + b3(σλ′~u2)x + b4,
ax = a1~u2 + a2~u2,t + a3(σλ′~u2)x + a4,
~u0,tt + σλb~u0,xt − σ2λ2a~u0,xx = ~f (5.8)
with the initial conditions
~u1(x, 0) = ~ϕ1(x), ~u1,t(x, 0) = ~ϕ2(x),
~u2(x, 0) = ~ϕ0,x(x), ~u2,t(x, 0) = ~ϕ1,x(x),
~u0(x, 0) = ~ϕ0(x), ~u0,t(x, 0) = ~ϕ1(x).
The functions b, a, bt, at depend on (~u0, ~u1, σλ′~ux, x, t) and f, ft, fj depend on(~u0, ~u0,t, σλ′~u2, x, t). The vector ~ϕ2 is defined similar to ϕ2 in assumption A 8.
Proof We consider only the ⇐–direction.
We differentiate (5.8) with respect to x, subtract the equation from (5.7) and obtain
(~u2 − ~u0,tt) + σλb(~u2 − ~u0,xt)xt − σ2λ2a(~u2 − ~u0,x)xx
+ σ′λb(~u2 − ~u0,x)t + σλ(b1~u2 + b2~u2,t + b3(σλ′~u2)x + b4)~u2,t
− σλ(b1~u0,x + b2~u1,x + b3(σλ′~u2)x + b4)~u0,xt
− (σ2)′λ2a(~u2 − ~u0,x)x − σ2λ2(a1~u2 + a2~u2,t + a3(σλ′~u2)x + a4)~u2,x
+ σ2λ2(a1~u0,x + a2~u1,x + a3(σλ′~u2)x + a4)~u0,xx
= ~f1(~u2 − ~u0,x) + ~f2(~u2 − ~u0,x)t. (5.9)
30 6 EXAMPLES
Differentiating (5.8) with respect to t and subtraction from (5.6) implies
(~u1 − ~u0,t)tt + σλb(~u1 − ~u0,t)xt − σ2λ2a(~u1 − ~u0,t)xx + σλ′b(~u2 − ~u0,x)t (5.10)
+ σλbt(~u2 − ~u0,x)t − σ2(λ2)′a(~u2 − ~u0,x)x − σ2λ2at(~u2 − ~u0,x)x = 0.
The equations (5.9), (5.10) are a weakly hyperbolic linear homogeneous system forthe functions ~v1 = ~u0,t − ~u1, ~v2 = ~u0,x − ~u2. We leave it to the reader to verify thatthe Levi conditions are satisfied. Hence one obtains ~v1 ≡ ~v2 ≡ 0. �
6 Examples
Example 1 The methods presented in this paper enable us to study equations with,e.g.,
λ(t) = tl exp
(
− 1
tα
)
, σ(x) = (sinx)k sin
(
1
sinx
)
,
where l ∈ Z, α ∈ R+, k ∈ N large. The degeneration occurs in the set
(R × {0}) ∪ ({mπ : m ∈ Z} × [0, T ])⋃
({
arcsin
(
1
mπ
)
: m ∈ Z\{0}}
× [0, T ]
)
The following example goes back to Qi Min–you [Qi 58].
Example 2 We consider
utt − t2uxx = aux, u(x, 0) = ϕ(x), ut(x, 0) = 0.
If a = const., a = 4n + 1, where n ≥ 0 is an integer, then the unique solution hasthe representation
u(x, t) =n∑
k=0
√πt2k
k!(n − k)!Γ(k + 12)
ϕ(k)
(
x +1
2t2)
.
One can observe two interesting effects. First, a loss of regularity occurs, whichdepends on the value of the coefficient of the lower order term. Second, singularitiesof the initial data propagate along the characteristic x + 1
2 t2 = const. Propagationalong the other characteristic x − 1
2 t2 = const. does not happen.
We will generalize this example.
Example 3 We consider the Cauchy problem
utt + btluxt − at2luxx − dtl−1ux = 0, (6.1)
u(x, 0) = ϕ(x), ut(x, 0) = 0, (6.2)
31
where a, b, d ∈ R, l ∈ N, l ≥ 1.
The ansatz u(x, t) =∑n
k=0 ckt(l+1)kϕ(k)
(
x + κtl+1)
leads to
c0 = 1, ck = −κ(l + 1)(2(l + 1)(k − 1) + l) + b(l + 1)(k − 1) − d
(l + 1)k((l + 1)k − 1)ck−1,
κ1,2 = − 1
2(l + 1)
(
b ∓√
b2 + 4a)
,
n =−l(l + 1)κ + d
2(l + 1)2κ + (l + 1)b.
This gives for κ = κ1 or κ = κ2
n1 =l
2(l + 1)
(
b√b2 + 4a
− 1
)
+d
(l + 1)√
b2 + 4a,
n2 = − l
2(l + 1)
(
b√b2 + 4a
+ 1
)
− d
(l + 1)√
b2 + 4a,
respectively. The natural number n describes the loss of Sobolev regularity.
It is worth to point out that x+κtl+1 = x+τ1,2Λ(t), where τ1,2 are the characteristic
roots and Λ(t) =∫ t
0 λ(s) ds, where λ(t) = tl. Additionally, we emphasize thatit is not possible to construct a solution as a linear combination of
∑n1j=0 . . . and
∑n2j=0 . . . , since n1 + n2 < 0.
The question arises of whether the solution suffers from a loss of generality if theexpressions for n1 and n2 are no integers, too. Here one can not express the solutionby the above finite sum. But there are other explicit representations. The followingideas go back to Taniguchi and Tozaki [TT80], who studied equations (6.1) withb = 0 and a = 1.
We apply partial Fourier transform with respect to x and obtain with y(t; ξ) =Fx→ξu(x, t) the ordinary differential equation with parameter ξ
y′′(t) + ibtlξy′(t) + at2lξ2y(t) − idtl−1ξy(t) = 0.
We transform the time variable,
τ := Λ(t)ξ, v(τ) := y(t; ξ)
and conclude that
v′′(τ) +
(
1
τ
l
l + 1+ ib
)
v′(τ) +
(
a − id1
l + 1
1
τ
)
v = 0.
We change the variables again,
z := riτ, w(z) := v( z
riτ
)
ezs ,
32 6 EXAMPLES
which gives
w′′(z) +
(
b
r− 2
s+
l
l + 1
1
z
)
w′(z)
+
(
1
s2− b
rs− a
r2− l
l + 1
1
sz− 1
l + 1
d
rz
)
w(z) = 0.
We choose1
s2− b
rs− a
r2= 0,
2
s− b
r= 1, |r| =
√
b2 + 4a.
The result is
zw′′(z) + (γ − z)w′(z) − αw(z) = 0,
γ =l
l + 1, α =
l
2(l + 1)
(
b√b2 + 4a
+ 1
)
+d
(l + 1)√
b2 + 4a.
This is a confluent hypergeometric differential equation. Two linearly independentsolutions are
w1(z) = 1F1(α, γ, z), w2(z) = z1−γ1F1(1 + α − γ, 2 − γ, z).
These series are polynomials if −α or α − γ are nonnegative integers, respectively.We have for |z| → ∞ the asymptotic expansion
1F1(α, γ, z)
Γ(γ)=
e±iπαz−α
Γ(γ − α)
(
R−1∑
n=0
(α)n(1 + α − γ)n
n!(−z)−n + O(|z|−R)
)
+ezzα−γ
Γ(α)
(
S−1∑
n=0
(γ − α)n(1 − α)n
n!z−n + O(|z|−S)
)
,
the upper sign been taken if −π2 < arg z < 3
2π, the lower sign if − 32π < arg z ≤ − 1
2π([AS84],13.5.1). Here one can see the special role of the exponents −α = n2 andα−γ = n1. At least one of this exponents is negative, since n1 +n2 = −γ < 0. Thisnegative exponent gives no loss of regularity.
After some computations one obtains
u(x, t) =1
2π
∫
R
eixξp0(t, ξ)ϕ(ξ) dξ,
where the function p0 has the asymptotic behaviour |ξ|−α or |ξ|α−γ for |ξ| → ∞.
Summary Let ϕ ∈ Hs(R), s ∈ R and n := max(−α, α − γ). Then there exists asolution u ∈ Hs−n(R).
Remark 6.1 We consider the Cauchy problem (0.2),
utt + bλ(t)uxt − cλ(t)2uxx − dλ′(t)ux = 0,
33
where b, d, c are real constants and b2 + 4c > 0. We define
Λ(t) :=
∫ t
0λ(τ) dτ, y := x − b
2Λ(t), v(y, t) := u(x, t)
and get
vtt −(
b2
4+ c
)
vyy −(
b
2+ d
)
vy = 0.
The transformation
y =: z
√
b2
4+ c, w(z, t) := v(y, t)
yields
wtt − λ(t)2wzz −b + 2d√b2 + 4c
λ′(t)wz = 0.
This reduction makes the results of Aleksandrian [Ale84] and Taniguchi–Toza-ki [TT80] applicable to equations which have a term uxt.
A Appendix
Proof of Lemma 3.1 We use Lemma A.1 and get
EN (∂(h)j v) = EN (vt − βj(x, t, h)σvx)
≤ ‖vt‖HN (P ) + Cprod,N ‖βj(x, t, h)‖HN (P ) ‖σvx‖HN (P )
≤ C(1 + ‖h‖HN (P ))SN (v).
Similarly, we conclude that
‖σvx‖HN (P ) =
∥
∥
∥
∥
∥
∥
(
∂(h)1 − ∂
(h)2
)
v
β(h)1 − β
(h)2
∥
∥
∥
∥
∥
∥
HN (P )
≤ Cprod,N
∥
∥
∥
∥
∥
1
β(h)1 − β
(h)2
∥
∥
∥
∥
∥
HN (P )
E (h)N (v)
≤ C(1 + ‖h‖HN (P ))E(h)N (v),
‖vt‖HN (P ) ≤∥
∥
∥
∥
∥
β(h)2
β(h)2 − β
(h)1
∂(h)1 v
∥
∥
∥
∥
∥
HN (P )
+
∥
∥
∥
∥
∥
β(h)1
β(h)2 − β
(h)1
∂(h)2 v
∥
∥
∥
∥
∥
HN (P )
≤ C(1 + ‖h‖HN (P ))E(h)N (v). �
Lemma A.1 Let f ∈ CN (P ×K), where K ⊂ Rn and P ⊂ R are compact sets. Let
vi ∈ HNper(P ) with (x, v1(x), . . . , vn(x)) ∈ K for all x ∈ P .
(a) Then,
EN (f(., v1(.), . . . , vn(.))) ≤ ϕN (‖v1‖∞ , . . . , ‖vn‖∞) (EN (v1) + . . . EN (vn) + 1) .
34 A APPENDIX
(b) More precisely, if N ≥ 3, then
EN (f(., v1(.), . . . , vn(.)))
≤ ϕN (‖v1‖∞ , ‖v1,x‖∞ . . . , ‖vn‖∞ , ‖vn,x‖∞) (EN (v1) + . . . EN (vn−1) + EN−1(vn))
+∥
∥
∥f (0,...,0,1)(x, v1(x), . . . , vn(x))∥
∥
∥
∞
∥
∥∂Nx vn
∥
∥
2+
N∑
j=0
∥
∥
∥f (j,0,...,0)(x, v1(x), . . . , vn(x))∥
∥
∥
2.
Proof We make use of the Leibniz formula
djxf(x, v1(x), . . . , vn(x))
=∑
i+l=j
j!
i!
∑
l1+···+ln=l
l1∑
ν1=0
· · ·ln∑
νn=0
1
ν1! · · · νn!f (i,ν1,...,νn)(x, v1, . . . , vn)×
×n∏
k=1
∑
hk,1+...hk,νk=lk
hk,m≥1
(
∂hk,1x vk
)
· · ·(
∂hk,νkx vk
)
hk,1! · · · hk,νk!
where we use the following convention: If νk = 0, then lk = 0 and if all νk are 0,then i = j.
Another tool is the generalized Interpolation Inequality of Nirenberg–Gagliardo forperiodical functions,
∥
∥∂jxv∥
∥
r≤ C ‖∂m
x v‖j−nm−np ‖∂n
xv‖1− j−nm−n
q , n ≤ j ≤ m, (A.1)
1
r=
j − n
p(m − n)+
1
q
(
1 − j − n
m − n
)
.
By Holder’s Inequality,
∥
∥djxf(x, v1(x), . . . , vn(x))
∥
∥
2
≤ Cj
∑
i+l=j
∑
l1+···+ln=l
l1∑
ν1=0
· · ·ln∑
νn=0
∥
∥
∥f (i,ν1,...,νn)(x, v1, . . . , vn)∥
∥
∥
∞×
×n∏
k=1
∑
hk,1+...hk,νk=lk
hk,m≥1
∥
∥
∥
∥
(
∂hk,1x vk
)2∥
∥
∥
∥
12
αk,1
· · ·∥
∥
∥
∥
(
∂hk,νkx vk
)2∥
∥
∥
∥
12
αk,νk
,
wheren∑
k=1
νk∑
s=1
1
αk,s
≤ 1.
35
We choose αk,s = lhk,s
and apply (A.1) with n = 0, r = 2αk,s, p = 2, q = ∞, m = l,
j = hk,s. The result is
∥
∥
∥∂
hk,sx vk
∥
∥
∥
2αk,s
≤ C∥
∥
∥∂l
xvk
∥
∥
∥
hk,sl
2‖vk‖
1−hk,s
l∞ ,
thus∥
∥djxf(x, v1(x), . . . , vn(x))
∥
∥
2
≤ ˜ϕN (‖v1‖∞ , . . . , ‖vn‖∞)
∑
i+l=j
∑
l1+···+ln=l
n∏
k=1
∥
∥
∥∂l
xvk
∥
∥
∥
lkl
2+ 1
≤ ϕN (‖v1‖∞ , . . . , ‖vn‖∞)(
‖v1‖HN (P ) + · · · + ‖vn‖HN (P ) + 1)
.
Proof of (b) We use the finer estimate∥
∥dNx f(x, v1(x), . . . , vn(x))
∥
∥
2
≤ Cj
∑
i+l=N
∑
l1+···+ln=l
l1∑
ν1=0
· · ·ln∑
νn=0
◦∥
∥
∥f (i,ν1,...,νn)(x, v1, . . . , vn)∥
∥
∥
∞×
×n∏
k=1
∑
hk,1+...hk,νk=lk
hk,m≥1
∥
∥
∥
∥
(
∂hk,1x vk
)2∥
∥
∥
∥
12
αk,1
· · ·∥
∥
∥
∥
(
∂hk,νkx vk
)2∥
∥
∥
∥
12
αk,νk
+∥
∥
∥f (0,...,0,1)
∥
∥
∥
∞
∥
∥∂Nx vn
∥
∥
2+∥
∥
∥f (N,0,...,0)
∥
∥
∥
2,
where the ”◦” means that terms with ln = l = N , νn = 1 or i = N do not occur.We apply (A.1) with
r = 2αk,s :=
{
2(l−2)hk,s−1 : k = n,2(l−1)hk,s−1 : k < n,
m =
{
l − 1 : k = n,
l : k < n,
and p = 2, q = ∞, j = hk,s, n = 1. The result is
∥
∥
∥∂hk,sx vk
∥
∥
∥
2αk,s
≤ C∥
∥
∥∂lxvk
∥
∥
∥
1αk,s
2‖vk,x‖
1− 1αk,s
∞ (k < n),
∥
∥
∥∂
hn,sx vn
∥
∥
∥
2αn,s
≤ C∥
∥
∥∂l−1
x vn
∥
∥
∥
1αn,s
2‖vn,x‖
1− 1αn,s
∞ .
We have to check whether∑
k,s1
αk,s≤ 1. It holds
S :=n∑
k=1
νk∑
s=1
1
αk,s
=n−1∑
k=1
lk − νk
l − 1+
ln − νn
l − 2.
We have to distinguish three cases:
36 A APPENDIX
ln = l This gives l1 = · · · = ln−1 = 0 and νn ≥ 2, hence S ≤ 1.
1 ≤ ln ≤ l − 1 At least one νk (k < n) is positive, hence∑n−1
k=1 lk − νk ≤ l − ln − 1.It follows that
S ≤ l − ln − 1
l − 1+
ln − νn
l − 2≤ 1 − ln
l − 1+
ln − 1
l − 2≤ 1.
ln = 0 (trivial)
We conclude that
∥
∥dNx f(x, v1(x), . . . , vn(x))
∥
∥
2
≤ ˜ϕN (‖v1‖∞ , . . . , ‖vn‖∞)
∑
β∈B
n−1∏
k=1
∥
∥∂Nx vk
∥
∥
γβ,k
2‖vk,x‖
γ′
β,k∞
∥
∥∂N−1x vn
∥
∥
γβ,n
2‖vn,x‖
γ′
β,n∞
+∥
∥
∥f (0,...,0,1)
∥
∥
∥
∞
∥
∥∂Nx vn
∥
∥
2+
N∑
j=0
∥
∥
∥f (j,0,...,0)
∥
∥
∥
2,
where B is some index set. It holds∑n
k=1 γβ,k ≤ 1. If this sum is strictly less than1, we use the embeddings HN−1(P ) ⊂ L∞(P ) and HN−2(P ) ⊂ L∞(P ) to increasethe exponents of
∥
∥∂Nx vk
∥
∥
2and
∥
∥∂N−1x vn
∥
∥
2such that the sum becomes 1.
The application of Young’s Inequality completes the proof. �
Remark A.1 After obvious modifications one can prove for every 0 ≤ m < n thegeneralization
EN (f(., v1(.), . . . , vn(.))) ≤ ϕN (‖v1‖∞ , ‖v1,x‖∞ . . . , ‖vn‖∞ , ‖vn,x‖∞)
× (EN (v1) + · · · + EN (vm) + EN−1(vm+1) + · · · + EN−1(vn))
+
n∑
j=m
∥
∥
∥f (0,...,0,1,0,...,0)(x, v1(x), . . . , vn(x))∥
∥
∥
∞
∥
∥∂Nx vj
∥
∥
2
+
N∑
j=0
∥
∥
∥f (j,0,...,0)(x, v1(x), . . . , vn(x))
∥
∥
∥
2.
Lemma A.2 Let σ ∈ HN+1per (P ), a ∈ HN
per(P ) and σa ∈ HN+1per (P ), where N ≥ 2.
Let h ∈ C∞0 (R) be a function with
∫
R
h(x) dx = 1, supph = [−1, 1]
and write hε(x) := ε−1h(x/ε), aε := a ∗ hε. Then it holds
‖σaε‖HN+1(P ) ≤ C ∀ε.
REFERENCES 37
Proof It is sufficient to prove that∥
∥σ∂N+1x aε
∥
∥
2≤ C. It holds
σ(x)(
∂N+1x aε
)
(x) =
∫
R
σ(x)a(N)(z)h′ε(x − z) dz = fε(x) + gε(x)
:=
∫
R
σ(z)a(N)(z)h′ε(x − z) dz +
∫
R
(σ(x) − σ(z))a(N)(z)h′ε(x − z) dz.
The proof is divided into two parts. In the first part we show that ‖fε‖2 ≤ C, inthe second we prove that ‖gε‖2 ≤ C. We have
fε(x) =
∫
R
(σa)(N)(z)h′ε(x − z)dz −
N∑
j=1
(
N
j
)∫
R
(σ(j)a(N−j))(z)h′ε(x − z)dz.
We observe that (σa)(N) ∈ H1per(P ), σ(j)a(N−j) ∈ H1
per(P ), which results in
fε(x) = (σa)(N+1) ∗ hε(x) −N∑
j=1
(
N
j
)
(σ(j)a(N−j))′ ∗ hε(x).
This implies immediately ‖fε‖2 ≤ C for all ε. Now we consider gε(x). Obviously,we have |σ(x)− σ(z)| ≤ C|x− z| ≤ Cε and |h′
ε(x− z)| ≤ Cε−2. This gives |gε(x)| ≤C(|a(N)| ∗ kε)(x), where
kε(x) =
{
(2ε)−1 : |x| ≤ ε,
0 : |x| > ε,‖kε‖L1(R) = 1.
By standard arguments one shows that
‖gε‖L2(P ) ≤ C∥
∥
∥|a(N)|
∥
∥
∥
L2(3P )‖kε‖L1(R) ≤ C
∥
∥
∥a(N)
∥
∥
∥
L2(P )≤ C. �
Acknowledgement: The research of the authors was supported by the project322-vigoni-dr.
References
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Michael DreherFaculty of Mathematics and Computer SciencesTechnical University Bergakademie FreibergBernhard von Cotta Str. 209596 Freiberg, GermanyE-mail: [email protected]
REFERENCES 39
Michael ReissigFaculty of Mathematics and Computer SciencesTechnical University Bergakademie FreibergBernhard von Cotta Str. 209596 Freiberg, GermanyE-Mail: [email protected]
